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FIGURE 2.1 Types of strain. (a) Tensile. (b) Compressive. (c) Shear. All deformation processes in manufacturing involve strains of these types. Tensile strains are involved in stretching sheet metal to make car bodies, compressive strains in forging metals to make turbine disks, and shear strains in making holes by punching.
FIGURE 2.2 (a) Original and final shape of a standard tensile-test specimen. (b) Outline of a tensile-test sequence showing different stages in the elongation of the specimen.
FIGURE 2.3 Schematic illustration of loading and unloading of a tensile-test specimen. Note that during unloading the curve follows a path parallel to the original elastic slope.
Elastic recovery
Permanentdeformation
Strain
Unload
Load
Str
ess
FIGURE 2.4 Total elongation in a tensile test as a function of original gage length for various metals. Because necking is a local phenomenon, elongation decreases with gage length. Standard gage length is usually 2 in. (50 mm), although shorter ones can be used if larger specimens are not available.
FIGURE 2.5 (a) True stress--true strain curve in tension. Note that, unlike in an engineering stress-strain curve, the slope is always positive and that the slope decreases with increasing strain. Although in the elastic range stress and strain are proportional, the total curve can be approximated by the power expression shown. On this curve, Y is the yield stress and Yf is the flow stress. (b) True stress-true strain curve plotted on a log-log scale. (c) True stress-true strain curve in tension for 1100-O aluminum plotted on a log-log scale. Note the large difference in the slopes in the elastic and plastic ranges. Source: After R. M. Caddell and R. Sowerby.
FIGURE 2.6 True stress-true strain curves in tension at room temperature for various metals. The point of intersection of each curve at the ordinate is the yield stress Y; thus, the elastic portions of the curves are not indicated. When the K and n values are determined from these curves, they may not agree with those given in Table 2.3 because of the different sources from which they were collected. Source: S. Kalpakjian.
FIGURE 2.7 Schematic illustration of various types of idealized stress-strain curves. (a) Perfectly elastic. (b) Rigid, perfectly plastic. (c) Elastic, perfectly plastic. (d) Rigid, linearly strain hardening. (e) Elastic, linearly strain hardening. The broken lines and arrows indicate unloading and reloading during the test.
(b) (c)
=Y + Ep!
(e)
Y
Y + Ep( -Y/E )
Y
(d)
(a)
Y= Y
tan-1 E
tan-1 Ep
Strain
Str
ess = EP
Y= Y
Y/E
= E
Y/E
E
FIGURE 2.8 The effect of strain-hardening exponent n on the shape of true stress-true strain curves. When n = 1, the material is elastic, and when n = 0, it is rigid and perfectly plastic.
FIGURE 2.9 Effect of temperature on mechanical properties of a carbon steel. Most materials display similar temperature sensitivity for elastic modulus, yield strength, ultimate strength, and ductility.
0 200 400 600
(°C)
0 200 400 600 800 1000 1200 1400
Temperature (°F)
Str
ess (p
si 3 1
03)
120
80
40
0
Str
ess (
MP
a)
600
400
200
0
Elongation
Elastic modulus
200
150
100
50
0
Ela
stic m
odulu
s (
GP
a)
Elo
ngation (
%)
0
20
40
60
Tensile strength
Yield strength
200
100
50
10
1
2
4
6
8
10
20
30
40
10-6 10-4 10-2 100 102 104 106
Strain rate (s-1)
Tensile
str
ength
(psi x 1
03)
800°
600°
400°
200°
30°C
MP
a
1000
°
FIGURE 2.10 The effect of strain rate on the ultimate tensile strength of aluminum. Note that as temperature increases, the slope increases. Thus, tensile strength becomes more and more sensitive to strain rate as temperature increases. Source: After J. H. Hollomon.
FIGURE 2.11 Dependence of the strain-rate sensitivity exponent m on the homologous temperature T/Tm for various materials. T is the testing temperature and Tm is the melting point of the metal, both on the absolute scale. The transition in the slopes of the curve occurs at about the recrystallization temperature of the metals. Source: After F. W. Boulger.
Titanium 200-1000 135-2 930-14 0.04-0.3Titanium alloys 200-1000 130-5 900-35 0.02-0.3Ti-6Al-4V" 815-930 9.5-1.6 65-11 0.50-0.80Zirconium 200-1000 120-4 830-27 0.04-0.4" at a strain rate of 2! 10#4 s#1.Note: As temperature increases, C decreases and m increases. As strainincreases, C increases and m may increase or decrease, or it may becomenegative within certain ranges of temperature and strain.Source: After T. Altan and F.W. Boulger.
TABLE 2.5 Approximate range of values for C and m in Eq. (2.16) for various annealed metals at true strains ranging from 0.2 to 1.0.
FIGURE 2.12 (a) The effect of strain-rate sensitivity exponent m on the total elongation for various metals. Note that elongation at high values of m approaches 1000%. Source: After D. Lee and W.A. Backofen. (b) The effect of strain-rate sensitivity exponent m on the post uniform (after necking) elongation for various metals. Source: After A.K. Ghosh.
0.20.1 0.3 0.4 0.5 0.6 0.7010
102
103
To
tal e
lon
ga
tio
n (
%)
m
Ti–5A1–2.5SnTi–6A1–4VZircaloy–4High- and ultra-high-carbon steels
FIGURE 2.13 The effect of hydrostatic pressure on true strain at fracture in tension for various metals. Even cast iron becomes ductile under high pressure. Source: After H.L.D. Pugh and D. Green.
5
4
3
2
1
0 300 600 900
MPa
00 20 40 60 80 100 120
Tru
e s
train
at fr
actu
re (
f)
Zinc
Copper
Aluminum
MagnesiumCast iron
Hydrostatic pressure (ksi)
FIGURE 2.14 Barreling in compressing a round solid cylindrical specimen (7075-O aluminum) between flat dies. Barreling is caused by friction at the die-specimen interfaces, which retards the free flow of the material. See also Figs.6.1 and 6.2. Source: K.M. Kulkarni and S. Kalpakjian.
FIGURE 2.15 Schematic illustration of the plane-strain compression test. The dimensional relationships shown should be satisfied for this test to be useful and reproducible. This test gives the yield stress of the material in plane strain, Y’. Source: After A. Nadai and H. Ford.
FIGURE 2.16 True stress-true strain curve in tension and compression for aluminum. For ductile metals, the curves for tension and compression are identical. Source: After A.H. Cottrell.
TensionCompression
0
4
8
12
16
Tru
e s
tress (
ksi)
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
True strain ( )
MP
a
100
50
0
FIGURE 2.17 Schematic illustration of the Bauschinger effect. Arrows show loading and unloading paths. Note the decrease in the yield stress in compression after the specimen has been subjected to tension. The same result is obtained if compression is applied first, followed by tension, whereby the yield stress in tension decreases.
FIGURE 2.18 Disk test on a brittle material, showing the direction of loading and the fracture path. This test is useful for brittle materials, such as ceramics and carbides.
Fracture
P
P
σ=2Pπdt
FIGURE 2.19 A typical torsion-test specimen. It is mounted between the two heads of a machine and is twisted. Note the shear deformation of an element in the reduced section.
FIGURE 2.21 Two bend-test methods for brittle materials: (a) three-point bending; (b) four-point bending. The shaded areas on the beams represent the bending-moment diagrams, described in texts on the mechanics of solids. Note the region of constant maximum bending moment in (b), whereas the maximum bending moment occurs only at the center of the specimen in (a).
FIGURE 2.22 General characteristics of hardness testing methods. The Knoop test is known as a microhardness test because of the light load and small impressions. Source: After H.W. Hayden, W.G. Moffatt, and V. Wulff.
FIGURE 2.23 Indentation geometry for Brinell hardness testing: (a) annealed metal; (b) work-hardened metal. Note the difference in metal flow at the periphery of the impressions.
(a) (b)
d
d
FIGURE 2.25 Bulk deformation in mild steel under a spherical indenter. Note that the depth of the deformed zone is about one order of magnitude larger than the depth of indentation. For a hardness test to be valid, the material should be allowed to fully develop this zone. This is why thinner specimens require smaller indentations. Source: Courtesy of M.C. Shaw and C.T. Yang.
FIGURE 2.24 Relation between Brinell hardness and yield stress for aluminum and steels. For comparison, the Brinell hardness (which is always measured in kg/mm2) is converted to psi units on the left scale.
FIGURE 2.28 Schematic illustration of a typical creep curve. The linear segment of the curve (constant slope) is useful in designing components for a specific creep life.
Primary
Time
Tertiary
Strain
Instantaneousdeformation
Rupture
Secondary
Specimen(10 x 10 x 75 mm)
Pendulum
Pendulum
Notch
Specimen(10 x 10 x 55 mm)
(a) (b)
FIGURE 2.29 Impact test specimens: (a) Charpy; (b) lzod.
FIGURE 2.30 Residual stresses developed in bending a beam made of an elastic, strain-hardening material. Note that unloading is equivalent to applying an equal and opposite moment to the part, as shown in (b). Because of nonuniform deformation, most parts made by plastic deformation processes contain residual stresses. Note that the forces and moments due to residual stresses must be internally balanced.
FIGURE 2.31 Distortion of parts with residual stresses after cutting or slitting: (a) rolled sheet or plate; (b) drawn rod; (c) thin-walled tubing. Because of the presence of residual stresses on the surfaces of parts, a round drill may produce an oval-shaped hole because of relaxation of stresses when a portion is removed.
FIGURE 2.32 Elimination of residual stresses by stretching. Residual stresses can be also reduced or eliminated by thermal treatments, such as stress relieving or annealing.
FIGURE 2.33 The state of stress in various metalworking operations. (a) Expansion of a thin-walled spherical shell under internal pressure. (b) Drawing of round rod or wire through a conical die to reduce its diameter; see Section 6.5 (c) Deep drawing of sheet metal with a punch and die to make a cup; see Section 7.6.
FIGURE 2.35 Examples of states of stress: (a) plane stress in sheet stretching; there are no stresses acting on the surfaces of the sheet. (b) plane stress in compression; there are no stresses acting on the sides of the specimen being compressed. (c) plane strain in tension; the width of the sheet remains constant while being stretched. (d) plane strain in compression (see also Fig. 2.15); the width of the specimen remains constant due to the restraint by the groove. (e) Triaxial tensile stresses acting on an element. (f) Hydrostatic compression of an element. Note also that an element on the cylindrical portion of a thin-walled tube in torsion is in the condition of both plane stress and plane strain (see also Section 2.11.7).
FIGURE 2.37 Schematic illustration of true stress-true strain curve showing yield stress Y, average flow stress, specific energy u1 and flow stress Yf.
FIGURE 2.38 Deformation of grid patterns in a workpiece: (a) original pattern; (b) after ideal deformation; (c) after inhomogeneous deformation, requiring redundant work of deformation. Note that (c) is basically (b) with additional shearing, especially at the outer layers. Thus (c) requires greater work of deformation than (b). See also Figs. 6.3 and 6.49.